Page 206 - Dynamic Loading and Design of Structures

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atmospheric pressure. This is expressed by use of Bernoulli’s equation:

(5.2e)

It is noted that the free surface conditions are non-linear, and that they are to be fulfilled on a
free surface which is not known until the problem is solved.

Linear theory
The Airy theory is based on a linearization (i.e. Φis supposed to be proportional to the wave

amplitude), the wave elevation amplitude ζis small (i.e. derivatives of ζare zero) and the

a
velocity square terms in eqn (5.2e) are neglected. This also means that the free surface

boundary conditions can be satisfied on z=0 instead of z=ζ
.
The solution of the linearized problem, as obtained by separation of variables (e.g. Clauss
et al., 1991), may be written as

(5.3)

with the circular frequency:

(5.4)

and the wave number:

(5.5)

The wave elevation is given by:

(5.6)

Equation (5.6) represents a wave propagating along the positive x-axis. The linearized
dynamic pressure is

(5.7)

and the velocities and accelerations in the x and z directions are: